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On the Operator Identity ∑ AkXBk ≡ 0

Published online by Cambridge University Press:  20 November 2018

C. K. Fong  and
A. R. Sourour
C. K. Fong
Affiliation:
University of Toronto, Toronto, Ontario
A. R. Sourour
Affiliation:
University of Victoria, Victoria, British Columbia
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Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by

(1)

We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form

For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 845 - 857
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Anderson, J. and Foias, C., Properties which normal operators share with normal derivations and related operators, Pac. J. Math.. 61 (1975), 313–325.Google Scholar
2. Calkin, J. W., Two sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math.. 42 (1941), 839–873.Google Scholar
3. Davis, C. and Rosenthal, P., Solving linear operator equations, Can. J. Math.. 26 (1974), 1384–1389.Google Scholar
4. Dunford, N. and Schwartz, J. T., Linear operators. Part I, (Interscience, New York, 1958).Google Scholar
5. Ho, Y., A note on derivations, Bull. Inst. Math. Acad. Sinica. 5 (1977), 1–5.Google Scholar
6. Lumer, G. and Rosenblum, M., Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41.Google Scholar
7. Rosenblum, M., On the operator equation BX- XA = Q, Duke Math. J. 23 (1956), 263–269.Google Scholar
8. Sinclair, A. M., Jordan homomorphisms and derivations on semi-simple Banach algebra, Proc. Amer. Math. Soc. 24 (1970), 209–214.Google Scholar
9. \Tala, K., On compact sets of compact operators, Ann. Acad. Sci. Fenn. AI No. 351 (1964).Google Scholar
10. Voiculescu, D., A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl.. 21 (1976), 97–113.Google Scholar
11. Williams, J. P., On the range of a derivation, Pac. J. Math.. 38 (1971), 273–279.Google Scholar